Av(1342, 2143, 2341, 3412)
View Raw Data
Generating Function
\(\displaystyle \frac{x^{8}-x^{7}-13 x^{6}+37 x^{5}-63 x^{4}+62 x^{3}-33 x^{2}+9 x -1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{3}}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 20, 62, 176, 469, 1201, 3005, 7428, 18270, 44920, 110721, 274073, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle -\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+x^{8}-x^{7}-13 x^{6}+37 x^{5}-63 x^{4}+62 x^{3}-33 x^{2}+9 x -1 = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 62\)
\(\displaystyle a \! \left(6\right) = 176\)
\(\displaystyle a \! \left(7\right) = 469\)
\(\displaystyle a \! \left(8\right) = 1201\)
\(\displaystyle a \! \left(n +4\right) = n^{2}-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-4 a \! \left(n \right)+16 a \! \left(n +1\right)-4 n -2, \quad n \geq 9\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-4 \sqrt{5}+20\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{40}+\frac{\left(4 \sqrt{5}+20\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{40}-n^{2}+\frac{3 \,2^{n} n}{8}+\\2 n -2^{-1+n}-1 & \text{otherwise} \end{array}\right.\)
Copy to clipboard:

This specification was found using the strategy pack "Requirement Placements" and has 83 rules.

Found on July 23, 2021.

Finding the specification took 13 seconds.

This tree is too big to show here. Click to view tree on new page.

Copy 83 equations to clipboard:
\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{6}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= x\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{14}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{32}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{43}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{41}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{62}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{42}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{56}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{59}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{60}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{62}\! \left(x \right)+F_{64}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{54}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{6}\! \left(x \right) F_{72}\! \left(x \right) F_{73}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{14}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{14}\! \left(x \right) F_{79}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{77}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{6}\! \left(x \right) F_{80}\! \left(x \right) F_{9}\! \left(x \right)\\ \end{align*}\)